Week 5: Addition & Subtraction with Different Bases

This week we only had class on Tuesday the 5th and our first homework assignment was due. We started off Tuesday working on some addition and subtraction practice problems using multiple techniques to solve them.

Our first practice problem was 64 + 77. Here are two examples on how to solve this equation using two different techniques, Partial Sums and Give and Take.


Partial Sums Visual:

Give and Take Visual:












For our second practice problem we had the subtraction problem 145 - 66. The two examples to solve this problems is using the techniques of Compensating and Holy Shift.

Compensating Visual:

Holy Shift Visual:

 

 



We also looked at a time problem to interlude into to our main topic of adding and subtraction with different bases.

 

 

Time Addition Visual:

 

Time Addition Graphed:

 

 

 

The last portion of the class we looked at addition and subtraction with different bases. In elementary school we learned to add in base 10 with stacking and borrowing. This exercise uses simple addition but with different bases. This activity's main goal was for us as future teachers to understand the frustration and confusion elementary students feel when learning stacking and borrowing and to apply that to our teaching methods.

Addition in Different Bases Visuals:

 

Example 1:

Example 2:

 

 

Subtraction in Different Bases Visuals:

 

Example 1:

 

Example 2:

 

 

 This will be my last blog post thanks for all the comments and go Cougs!

Week 4: Subtraction and Addition

On Tuesday the 29th we worked out of our classroom activities booklet on pages 51-55 (handwritten) using 100s, 10s, and 1s number blocks, index cards, dice, and manipulatives.

Since the number blocks and the index cards were already created we worked on activities 2 through 5.

In activity 2 we used dice to determine the amount of base ten blocks we would use to make two three-digit numbers. After rolling the dice we would create the number by using the correct amount number of blocks. If it was possible your group could exchange the amount of 1s, 10s, or 100s to make an easier total of blocks (similar to exchanging money). Once the numbers were established in their simplest form we wrote a number sentence to compare the two words. An example would be 445 < 569.

In activity 3 we used the same base 10 blocks to do addition. First we picked out two different three-digit numbers and represented them in the blocks. After forming the numbers in blocks we combined the numbers by combining the blocks and exchanging them to create a simple answer. All the steps and diagrams were recorded on a piece of paper we would later turn in.

In activity 4 we did a more abstract activity by using single manipulative pieces to represent numbers instead of a more concrete form with the base 10 blocks. Using the index cards to create the two three-digit numbers we expressed them by placing the correct number on of manipulatives onto a place value mat. Once placing the correct amount we exchanged the manipulatives to create the sum.

 

 

 

In activity 5 we did the same process of creating two three-digit numbers and adding them together but this didn't involve any form of manipulatives by using two different addition techniques; Addition without Regrouping and addition with regrouping (also known as stacking and borrowing).

Addition without Regrouping:

Addition with regrouping:

 

On Thursday the 31st we looked at solving an addition and a subtraction problem using multiple techniques to solve each.

 

 

Our first problem was an addition problem, 34 + 28 = 62. Here are five examples to solve this problem.

 

 

Traditional:

Give & Take:

 

 



 

Partial Sums / Instructional Algorithm:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Compensating:



 

Decomposing:

 

Our second problem was a subtraction problem 52 - 37 = 15. There are six examples to solve this problem.

 

Traditional:



 

Partial Differences:

 

Compensating:

 

 

Distance:

 

Decomposing:

 

Holy Shift:

 These techniques are important to remember for set 3 in our homework which is due soon. Stay tuned for next week's blog and go Cougs!

 

 



Week 3: Addition & Subtraction

Class resumed on Tuesday of this week and we continued doing some labeling practice for sets. Some of these practice problems involved shading with parentheses. Here are some of the problems we worked on in class.

1.
Shorthand notation:

 

 

 

Visual:

 

2.

Shorthand notation:

 

 



Visual:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3.

Shorthand notation

 

Visual:

 

4.

Shorthand notation:

 

Visual:

 

 

 



For more practice you can use http://nlvm.usu.edu/en/nav/frames_asid_153_g_2_t_1.html?open=instructions&from=search.html?qt=venn .

 

On Thursday we started off working on sets, but instead of labeling that we have been doing, we incorporated variables such as letters found in two names.

 

5.



 

Visual:

 

In this problem the letters common letter in each set are underlined. When putting them into the Venn diagram only need to be written once.

In class we also looked at the properties of addition. The first two were the Set Model of Addition and the Measurement/Number Line/Active Model.

Here is an example of each to distinguish between the two when looking at a word problem.

Example of a Set:

Eroll has seven chocolate chip cookies and eight oatmeal cookies. How many does he have altogether?

Example of a Measurement:

Eroll has seven chocolate chip cookies and bakes eight more. How many does he have in total?

Other properties of addition are:

Commutative Property: A + B = B + A

In this property moving the numbers around in order to make the problem easier to solve.

Associative Property: A + (B + C) = (A + B) + C

Grouping numbers together in different ways

Identity Property of 0: A + 0 = A

When adding zero the number stays the same.

We also looked at Number Relationships in math. These concepts are the building block for students to continue learning math.

Spatial Relationships:

Recognizing how many numbers there are without counting by seeing the visual pattern.

One & Two More or Less:

This is not the ability to count on two or count two back from a number, but instead knowing which numbers are one more or two less than any given number.

Benchmarks of 5 & 10:

Since 10 plays such an important role in our number system (2 5s make 10). Students must know how numbers relate to five and ten.

Part-Part-Whole:

To conceptualize a number as being made up of two or more parts is the most important relationship to develop.

We lastly looked at Models of Subtraction:

Take Away Example:

Eroll has eight dollars. He spends five dollars for a movie ticket. How much money does he have now? (8 - 5 = __ )

Missing Addend Example:

Eroll had five dollars. He found some lying on the ground. Now he has eight dollars. How much money did he find? (5 + __ = 8)

Comparison Example:

Eroll has eight dollars. Kyle has five. How much more money does Eroll have then Kyle?

Number Line/Measurement/Distance Example:

Eroll hiked eight miles. Five were before lunch. How many miles did he hike after lunch?

All of the concepts learned in this week can be applied into the second set of our homework.



Stay tuned for next week's blog and go Cougs!

Week 2: Problem solving & Sets

On Tuesday we finished up the concept of problem solving . On of the problems we focused on during class was a word problem involving find the perimeter a hexagon tiles with 1 centimeter sides. In solving the problems we were challenged to do so with out simply counting and  to use alegrabric ways to find the perimeter as the number of tiles increased.

Starting total:

When looking for the perimeter of a set of 100 hexagons we, as a class, came up with 3 algrebraic ways to solve the problem.

In solving  for the perimeter this way we also reviewed the words of constant (which are highlighted in red in the photos) and variables (the H in the problem).

On Thursday we began learning about sets. In doing so we reviewed the terms; union, intersection, and complement. we went over the shorthand notation and a visual way to represent the set.

Union:
Shorthand notation:

Visual:

Intersection:

Shorthand notation: 

Visual:

Complement:

Shorthand notation:

Visual:

 Shorthand notation:

Visual:

Shorthand notation:

Visual:

We also did an activity in class that helped in practicing labeling and making sets with 2-3 rings and colored shapes called attribute pieces. This activity can be found in our problem solving and mathematical reasoning class activity book on pages 27-40.

In doing this activity we also looked at the difference between "and" & "or" in looking at a set.

Visual for And:

Visual for Or:

Lastly, we began we added a third ring and continued with labeling and making a visual.

Shorthand notation example:

Visual:

If the concept of set is confusing there is no worry on Tuesday of next week we are continuing working with sets. This concept is important to grasp and understand since the will be found in our homework for set 2.1.

Stay tuned for next posting and Go Cougs!